Design of a Planetary-Cyclo-Drive Speed Reducer
Cycloid Stage, Geometry, Element Analyses
Växjö,2012-05-30
15p
Mechanical Engineering/ 2MT00E
Handledare: Hans Hansson, SwePart Transmissions AB
Handledare: Professor Samir Khoshaba, Linnéuniversitetet, Institutuionen för teknik
Examinator: Izudin Dugic
Thesis nr: TEK
Författare : Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Summary
This project has been assigned by SwePart Transmissions AB. It is about calculation and dimensioning,
of the elements in a cycloid stage of a speed reducer. Their idea is to use the results from this project and
go into production of such reducer to cover another segment of the market. The company is interested in
supplying transmissions for robust systems and for various industrial purposes, where large ratios of
speed reduction are needed.
The company has given the necessary input data for the model. They have also provided a real cyclodrive reducer for further analyses. To obtain the dimensions and forming the geometry of the gears,
some parts of Professor Ognyan Alipiev’s Phd work have been used. Professor Alipiev is head of
“Theory of Mechanisms and Machines” department in University of Ruse “Angel Kunchev”, Bulgaria.
For the determination of forces on the elements, models and drawings has been used Solidworks (SW)
CAD software and S W simulation environment. The resultant calculation process can be used for
designing the geometry and determination of the properties regarding the cycloid reducer.
Keywords: cyclo drive, cycloid speed reducer, epicycloid reducer
Acknowledgements
This project has been carried through during the spring semester 2012 at the Linnaeus University and
with the assistance of SwePart Transmissions AB.
We would like to present our gratitude to the following:
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Professor Ognyan Alipiev (Angel Kunchev University, Faculty of Mechanical and
Manufacturing Engineering, Ruse) for providing us with all the necessary theoretical data and
also responding to the questions we had. Thank you!
Samir Khoshaba (Linnaeus University, School of Engineering, Växjö) for supervision of the
project and all the good times. Thank you!
Hans Hansson (SwePart Transmissions AB, Liatorp) for introducing the problem and providing
us with the necessary data and also his time. Thank you!
Assistant Filip Genovski (Linnaeus University, School of Engineering, Växjö) Thank you!
Our parents for the support. Thank you!
Li Yawei and Wu Yuanzhe, for the group work we did together on this project. Thank you!
The ERASMUS student exchange program. Thank you!
The Swedish education system. Thank you!
CONTENTS
1.INTRODUCTION………………………………………………………………………………………………………………………….…1
1.1Background………………………………………………………………………………………………………………………………………1
1.2 Task/Problem Description……………………………………………………………………………………………………………….6
1.3 Limitations……………………………………………………………………………………………………………………………..……….6
2. THEORY…………………………..……………………………………………………………………….……………..……..……………..7
2.1 Geometry…………………………………………………………………………………………………………………………….……......7
2.1.1 The epicycloid curve………………………………………………………………………………………………………….………...7
2.1.2. Geometry of modified epicycloid gears………………………………………………………………….……………………8
2.1.3. General concept of eccentric tool meshing…………………………………………………………………...…………….9
2.1.4. Basic dimensions of epicycloidal gears…………………………………………………………………………..….…..….14
2.1.5. Equations for teeth profile and curvature radius……………………………………………………………..….…....17
2.1.6 Forming of the epicycloidal gear transmission………………………………………………………………….…………22
2.1.7. Action line. Contact Points…………………………………………………………………………………………………………24
2.1.8 Distinctive Meshing Angles…………………………………………………………………………………………….........……26
2.1.9 Contact ratio……………………………………………………………………………………………………………………………...28
2.2. Cycloid gearbox design………………………………………………………………………………………….…………………….31
2.2.1 Forces and Force distribution in the cycloid gear…………………………………………………………………….….31
2.2.2. Designing the holes and the thickness of the cycloid discs…………………………………………………….…..33
3. METOD………………………………………………………………………………………….………………………………………..……37
3.1 Modeling in SolidWorks…………………………………………………………………………………………….…………..........37
3.2. Getting the results………………………………………………………………………………….…….………….……………..……37
3.3 Making Assembly and Simulation……………………………………………………………………………….………………….38
4. RESULTS…………………………………………………………………………………………………..……………………………..……40
4.1. Total ratio…………………………………………………………………………………..….…….…………….………………...…….40
4.2 SolidWorks Simulation Results………………………………….…….….……………………………..….………………..….40
4.2.1.Cycloid Gear Results….……….………….……….………….……….…….………….……………………….…………..………41
4.2.2 Input Shaft Results………………………………………………………………………………………………….………………….47
4.2.3. Output Shaft Results…………………………………………………………….….……………….…..…..……………………..53
4.2.4. Eccentric Shaft Results………………………….………………….……………………………….……….……………….….…60
5. ANALISYS……………………….…………………………………………………………………………….…….…………..…………..67
6. CONCLUSION……………………………………………………………………………………………..…………….……….….……70
7. REFERENCES……………………………………………………………………………..………………..……………..…………..…..71
8. APPENDIX…………………………………………………………………………………..………………..……………....…………….72
1. Introduction
The main aspects of the project are presented in this chapter. The background will give a general
overview of the basic features about the product. The main aspects of the task are described in the
problem description sub-chapter.
1.1. Background
The project has been performed in cooperation with SwePart Transmission AB. The company is a
leading supplier of spare parts and transmissions for vehicles and industry. They have an idea to design a
gearbox, that can replace a model of a planetary-cycloid gearbox, used in many industries such as robust
systems, mines, metallurgy, chemical, and textile and in situations where high speed reduction is
needed. The high ratio is provided because of the planetary and cycloid stages of speed reduction. The
project is divided in two parts carried out by different groups. In this paper part one is presented, tough
some data has been used from the other group’s calculations. SwePart are primarily interested to know
more about the general equations for calculating the properties of the second reduction stage (e.g.
cycloidal stage).
This paper shows the geometrical knowledge for producing the special geometry required to draft the
cycloid gear wheel. Calculated example based on the provided input data is presented (Appendix 1).
The cycloid reducer is a speed reducer (shown in Figure 1 below).
Figure 1.Cycloid reducer (http://www.ohiobelting.com/images/shimpo4.jpg)
It consists of four main components: an input and output shaft, cycloid discs and housing with internal
pins. The input shaft has two eccentric roller bearings (eccentrically mounted cams), fitting into the
eccentric roller bearing of the wheels (discs). This eccentricity causes the center of the disc to rotate in
the housing. The wheels’ (discs’) teeth are less than the housing’s pins, causing the reverse orbit rotation
within the housing. If a fixed point of the circumference of the cycloid wheel (disc) is traced, the
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resulting path will describe a curve called a hypocycloid. The rotary motion is converted via the disc
rolling over within the housing, which accommodates the pins. The ratio of the gearbox depends on the
relative dimensions of the disc and the housing.
Engineers use gear reducers in mechanized industry and/or industrial-automation applications. Over the
time two types of reducers became a desired standard, because of their advantages, naming: cycloid and
planetary speed reducers. Using high speed engines for general purposes and for precise machines
working with servomotors; these machines require in many cases the rotation speed to be reduced and
the torque to be increased. The smallest and most compact gear used to achieve this task is the cycloplanetary gear (CPG) shown on Figure 2.
Figure 2. Cyclo Planetary Gear (CPG)
As mentioned, the reducer unit consists of four basic parts: the high-speed input shaft, a single or
compound cycloid planetary gear, an eccentric cam (eccentric bearing) and a slow-speed output shaft. In
Figure 3 is shown the exploded view of the assembly. In a lot of cases the design and terminology of the
components can vary .On this figure are also shown the other parts like the shielding, the bearing,
spacer, the ring gear housing also the slow shaft (output shaft) and shaft rollers. The input shaft is
connected to the eccentric cam, through a key connection. This rigid connection creates the eccentric
cycloid gear rotation i.e. cycloid motion. This motion causes the discs to roll over the ring gear rollers,
mounted in the ring gear housing (also later referred to as pins). The output shaft is connected to the
cycloid disc via shaft rollers. From the figure it can be observed that the shaft rollers are situated into the
cycloid disc’s perimeter holes (look at Figure.2). The torque is transferred from the perimeter holes, as
the disc rotates to the output shaft. These holes are bigger than the slow shafts rollers; which causes the
output pins to move around in the holes. Steady rotation of the output shaft is achieved from the
wobbling movement of the disc. The direction of rotation of the output shaft is the opposite of the
direction of the input shaft.
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According to Darali drives (2012,homepage) and Sumitomo drives (2012,homepage) single stage
efficiency approaches 93%, double stage 86% and the reduction rate of a single stage reducer is up to
199:1 and double stage up to 7569:1.
Figure 3. Exploded View of a Cycloid Speed Reducer
(http://www.gearmotor.com.my/wp-content/uploads/2010/02/cyclo-drive-spare-parts.jpg)
Gear ratio is designated by this expression:
(Eq 1.)
Where i – reduction rate
– number of teeth(lobes) on the cycloid disc
– number of pins in the housing .
Both cycloid and planetary reducers can be implemented in any industry where servo, stepper or
variable frequency drives are used (D’Amico, 2012). Both have their advantages in comparison, the
author has made this brief Table1 shown below:
Table 1. Reducer advantages in comparison (www.machinedesign.com)
Benefits of planetary gearboxes
Benefits of cycloidal gearboxes
High torque density
Zero or very-low backlash stays relatively constant
during life of the application
Load distribution and sharing between planet gears Rolling instead of sliding contact
Smooth operation
Low wear, Quiet operation
High efficiency
Shock-load capacity
Low input inertia
Torsional stiffness
Low backlash
Flat, pancake design
Low cost
Ratios exceeding 200:1 in a compact size
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-RV-type cyclo-drive
So far the main principles for the cycloid mechanism have been explained, regarding the concept where
the eccentric bearing is located in the center of the discs. The RV type in particular is a plano-ecentric
reduction gear mechanism working on the same “cyclo” principle.
The reducer is shown in Figure 2 in Appendix 2, and also has the advantages, the concept offers. Note
that the designed reducer differs from the one shown in the figures. Although as mentioned the principle
of speed reduction and power transfer are the same.
The RV is a 2-stage speed reducer which includes: a spur gear stage with involute teeth and an epicyclic
gear reduction stage. The first stage has an input gear (sun gear) coupled with in this case three satellite
gears. They are connected to the crankshafts via splines. The first stage can be modified to change the
overall unit ratio.
The crankshafts are driven by the spur gears. On the crankshafts are mounted the eccentric bearings,
which cause the eccentric (cycloidal) motions of the two discs. They are offset 180 degrees from one
another for equally balancing the load. The cycloid gear, driven by the eccentric shafts engages with the
pins in the case. The transferred power then is directed to the output shaft. The gear rotates the opposite
direction comparing to the input shaft. In fig.4 is shown how this works.
Figure 4. RV Principle (www.nabtesco.com)
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The mechanism block diagram below in Figure 5 shows the whole reducer and its components. On the
right side is shown the kinematic scheme, and on the left the cycloid stage.
Figure 5. Mechanism Block Diagram (www.nabtesco.com)
The overall reduction ratio of the reducer can be determined from the following equation:
(Eq 1.1)
Where here - is the reduction ratio, - is the number of teeth on the sun gear,
– number of teeth on
the satellite gears,
– number of pins in case.
In Figure 3 in Appendix 2 are shown some examples where these reducers are often used.
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1.3 Task/Problem Description
The main aim of the project is to design a planetary-cyclo-drive speed reducer. SwePart Company is
interested in a certain type of these reducers, consisting of two stages: a planetary and cycloid reduction
stages. Our assignment is to design the cycloid stage of the reducer. This includes generating the
equations for the geometry of a cycloid gear and analysis of the parts related to the motion (shafts :
input, output and eccentric bearing shaft) also dimensioning and testing the epicycloid gear. The other
group designed the shafts, the shaft related parts, in this paper have their designs have been tested and
evaluated in Solidworks simulation. Input data (rotational speed and torque) has been received from the
other group.
The work in this paper is divided in two main parts:
- Geometry
- Design
Geometry part consists of generating equations for the teeth profile, equations for the specific features
(number of teeth, modification, module, radius of the pins) and equations for the basic diameters.
Design part is about creating the real reducer with the input data given from the company. It is shown
what are the criteria’s that the parts should meet in order to function properly. This paper proposes a
simple and exact approach for the profile design of the cycloid gear and gearing, as the main part of the
reducer.
Design example is shown with the input data given from SwePart AB:
Ratio:
Output Torque:
Frequency of Rotation:
Safety factor (SF) ≥ 2
Material: Case hardened steel 20MnCr6
1.4 Limitations
Because of the time, needed to go further and deeper into the subject the paper shows only the main
concepts. Therefore only static analyses have been made. Also a big issue was finding the right
terminology for these transmissions in English since they are not standardized. Another limitation is also
the computers that were used, because the simulations take a lot of time.
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2. Theory
In this chapter are explained the main equations for the geometry of cycloid gears and transmissions.
The chapter is focused on epicycloid formation, main dimensions and teeth/lobe profiles of the gears
and transmission parameters.
2.1 Geometry
2.1.1The epicycloid curve
Symbol
Explanation
Radius of the epicycle
Epicycloids epicycle
Pins radius
Angle of rotation of the epicycle
Pitch circle radius
Dimension
mm
mm
mm
mm
mm
Table 2. Used symbols
Note
The definition for an epicycloid is - a curve, produced by tracing the path, generated by a chosen point
of a selected circle, called epicycle. This is rolling without slipping around another fixed circle.
There are three types of epicycloids - normal, shortened and extended (not shown because it has no
practical value here). On Figure 6 (a) can be observed the normal, on (b) the shortened epicycloid curve.
Here will be explained how the path is formed.
Looking at Figure 6 (a) – to generate the profile of the normal epicycloid, point
is rigidly connected
on the circle with radii . The circle will be rolling without slipping around the guiding circle with
radius .In this case it is shown that:
.
(a)
(b)
Figure 6. Curve generating (Alipiev, 1988 )
In Figure 6(b) can be seen a shortened epicycloid, the curve is also called in mathematics an epitrihoid.
The geometry is produced the same way as a normal epicycloid. The difference here is that the point
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lies in the perimeter of the rolling circle. Here can be written that for a epicycloid to be shortened it has
to satisfy the following:
.
All epicycloid gears have a theoretical and working profile. The theoretical profile is needed to help
define the actual working profile. The epicycloid (normal or shortened) is the theoretical profile, and the
working profile is an equidistant curve. To obtain the working profile a circle with radii (shown in
Figure 8) is presented. Its centre is related to point
. Here point K is the contact point between the
working profile and a circle with radii . Generating the working profile here has been significantly
simplified because of the use of Solidworks CAD environment.
2.1.2 Geometry of modified epicycloid gears
Symbol
Explanation
Table 3.Used symbols
Note
Dimension
mm
mm
mm
mm
Center distance
Radius of the pitch circle
Radius of the centrode of the epicycloid
Radius of the centrode of the pins circle
The cycloid drive shown in Figure7, consists of a epicycloid gear 1 with teeth (also called lobes) and 2
internal housing with pins. The profile itself has the shape of an equidistant curve (later referred to as
working profile) to the epicycloid curve (normal or shortened).
Figure7. Corrected cycloid drive(Alipiev, 1988)
Here is observed a transmission with difference in the number of teeth
and the gear equal to one, expressed by:
and
, between the housing
The centrodes (circles with common centers) of the epicycloid gear and housing are circles with radii
naming
and
. Making distinguish in function of the location of the centrode to the tooth profile,
corrected and not corrected gears will be recognized.
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With not addendum modified gears we will refer to those, which centrode circle of the epicycloid
). On the other
gearing is matching the circle over which the centers of the pins are located (
hand the case where
the gearing is modified. Not modified, epicycloid gearing with teeth
difference
is used in modern solutions. According to (Alipiev, 1988) it is better to use
modified. Actually modification is a necessity, as explained further in the paper. In this case the circle
on which the centers of the pins (of the housing) are located, is bigger than the centrode of the cycloid
). When the drives have difference in teeth equal to 1 - the modification is completely
gear (
necessary, meaning that
.
2.1.3. General concept of eccentric tool meshing
Symbol
p
m
X
x
Explanation
Angle of velocity of the instrument
Angle of velocity of the cycloid
Radius of the instrument
Length of the module circle step
Module
Coefficient of the radius of the profile generating circle
Center distance between module and profile generating circles
Modification
Coefficient of modification
Table 4.Used symbols
Dimension
Note
deg
deg
mm
mm
mm
mm
-
-Instrument meshing
In Figure 8 can be observed the eccentric tool meshing scheme. With O is marked the instrumental
wheel and with 1 the epicycloid gear/disc. The cutting instrument with radii is rotating with an offset
from its original axis. This creates an eccentric motion around . The cutting tool and the blank are
rotating in opposite directions. The proportion of the angular velocities determines the number of
lobes/teeth on the cycloid disc. With
is marked the velocity of the tool and with the velocity of the
disc, e.g. :
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Figure 8 Eccentric Tool Meshing (Alipiev, 1988)
Point meets the radius of the cutting instrument’s base circle and wheel’s: naming
and . This
point is referred to as tool meshing point. Circle
rotates without slipping over circle . On the other
hand point
lies in the perimeter of
. The same while rotating around
will generate the
theoretical profile, while the circle with radii will generate (cut) the working profile.
-Profile cutting instrument
One of the most important factors laying the foundations of modern technology is interchangeability.
Interchangeability could exist only if standardization is available. The gears are one of the most difficult
and accurate machine details to manufacture. Thus the standardization of some of the parameters of the
gears is the main task in order to produce accurate gears.
To cut the cycloid disc a special instrument is introduced. It has been called a profile generating contour
(shown on Figure 9, page 11).
When cutting an involute spur gear, a rack-cutter is used to generate the profile, but when cutting a
cycloid gear using a rack is not possible because of the following reasons:
1. If we use a rack-cutter, the generating profile would be a very complicated and nontechnological curve;
2. Using the rack we cannot express the displacement in any way.
On Figure 9 is introduced the module circle with radius and profile generating circle with radius .
The module circle is used as a base, so that the parameters of the profile cutting instrument and cycloid
gear have standard values. The module is actually the diameter of its module circle:
(Eq 2.1.1)
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The module m is measured in millimeters and its values are strongly regulated by standards.
The circumference p of the modular circle is called a pitch of the profile cutting instrument. The
module is determined by:
(Eq 2.1.2).
The radius of the profile generating circle is given with the following equation:
,
Where
radius
circle
(Eq 2.1.3)
is the coefficient of the circle radius. Its value is equal to 1, according to (Alipiev,1988). The
is equal to the radii of the pins of the cycloid housing and the center of the profile generating
is lying on the module circle.
e
Figure 9.Profile Generating Contour (Alipiev, 1988)
The distance between the centers of both, profile generating circle and module circle is designated by e
and is called eccentricity of the profile cutting instrument. It can be determined with the module
according to the following equation:
(Eq 2.1.4)
The dimensions of the profile cutting instrument can be determined by two independent from each other
parameters. The module witch acts like a scale factor, and coefficient of the profile generating circle
( . These parameters will directly determine the cycloid gears dimensions.
In particular the profile generating tool has an eccentric contour which has a determined pivot (Figure
10). In addition to the profile cutting instrument, with eccentricity
and radius
is added the
original circle. The eccentricity is equal to the distance from the pivot
and the center of the profile
generating circle . The original circle’s radius is equal to the radius of the modular circle (
), but in general they are not coincident.
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In summary the module circle and the profile generating circle as main components of the profile cutting
instrument do not change their mutual position. What changes is the location of the profile cutting
instrument over initial circle. This change is called addendum modification. The displacement is
measured from one pre-selected nominal position (Oo).
Figure 10.Profile cutting instrument (Alipiev, 1988)
-Addendum Modification
In Figure 11 can be seen the process of forming the teeth on the blank 1 by the teeth forming circle. The
resultant teeth are with determined geometric form. With
and
are designated respectively the
radius of the circles which belong to the cutting tool. As mentioned above
is equal to the radius of
the module circle,
is the radius of the pitch circle of the epicycloidal wheel. In all the four schemes in
Figure 11 the center distance is the same
. The only difference is the position of the
profile cutting instrument. The location of the profile cutting instrument affects the form, size and
disposal of the teeth.
equidistant
of normal
Epicycloid
equidistant
of extended
epicycloid
(a)
(b)
equidistant
of shortened
epicycloid
(c)
(d)
Figure 11. Forming Process (Alipiev, 1988)
The position of the profile cutting instrument in which the module circle coincides with the original
circle and simultaneously touches the pitch circle of the epicycloidal wheel is called nominal position
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(see Figure 11 (b)). Any other position is formed by displacing (modifying its position) the profile
cutting instrument. The epicycloidal wheel formed in a nominal position is called unmodified wheel.
If the profile cutting instrument is displaced from the nominal position in which the module circle is not
touching the pitch circle the result would be an unmodified wheel.
The modification of the profile cutting instrument is the difference between the eccentricity of the
profile cutting instrument and the eccentricity of the generating circle i.e.:
(Eq 2.1.5)
Apparently according to (Eq 2.1.5) the eccentricity could be positive, negative or nominal. If
then the displacement is positive (see Figure 11 (c)), if
then it is negative (look at Figure 11 (a))
and if
(the modification is equal to zero) the condition is called nominal. There is a fourth
condition X = e, the modification is equal to zero (
) and it is called a boundary condition (see
Figure 11 (d)).
The modification of the profile cutting instrument can be expressed by the following equation
(Eq 2.1.6)
Where x is the coefficient of modification. It can be either positive or negative, so the following
modifications are possible:
1.
2.
3.
4.
- without modification
– positive gears
– negative gears
– border gears
When looking at a transmission with a difference of teeth equal to one, practical value have only
positive modifications (see Figure 11 (c)). Also positive modification is a necessity otherwise the
tooth/lobe profile will result to be undercut.
From Figure 11 can be summarized the information so far: that all the basic parameters needed to define
the dimensions of the gear are:
a)
b)
c)
d)
The module – m;
Coefficient of the radius of the profile generating circle
;
Coefficient of modification of the profile cutting instrument - x;
Number of teeth of epicycloidal wheel .
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2.1.4. Basic dimensions of epicycloidal gears
Symbol
s
a
Explanation
Length of the pitch of the epicycloid wheel
Theoretical radius of the dedendum circle
Theoretical radius of the dedendum circle
Radius of the dedendum circle
Radius of the addendum
Tooth depth
Coefficient of the epicycloid shortening
Center distance
Dimension
mm
mm
mm
mm
mm
mm
mm
Table 5.Used symbols
Note
All the dimensions of the epicycloidal gears will be defined using Figure 1 in Appendix 2. All the basic
dimensions are derivatives of the basic parameters as it has been shown.
-
Step of the pitch circle
Pitch - according to the theory of mechanisms, it is the distance between two homonymous profiles. In
the process of cutting the epicycloidal gear the tool is rolling without slipping on the pitch circle. From
the general definition for pitch, follows that it is in this case the length of the enclosed curve divided by
the number of teeth.
Therefore, if we designate with p the pitch of the epicycloidal wheel on its pitch diameter and if we use
the relation
we can write down the following:
(Eq 2.1.7).
The module of the generating circle, which has formed a certain wheel is called module of epicycloidal
wheel.
- Pitch diameter
The circumference s of the pitch can be defined with the help of the following formulas:
(Eq 2.1.8)
(Eq 2.1.9)
If we substitute p from (Eq 2.1.7) and (Eq 2.1.8) in (Eq 2.1.9) and for the radius of the pitch diameter is
obtained the following equation:
(Eq 2.1.10)
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
For the diameter of the pitch circle we can write the following
(Eq2.1.11)
- Diameter of the theoretical dedendum circle
From Figure 1 in Appendix 2 can be observed that the section ̅̅̅̅̅ is equal to X of the profile cutting
instrument. If we take that into consideration for the radius of the theoretical dedendum we can write
down the following formula:
(Eq 2.1.12)
When we substitute (Eq 2.1.6) and (Eq 2.1.10) for the radius of the theoretical dedendum we receive:
(
- Diameter of theoretical addendum circle
According to the distances in Figure 1 in Appendix 2 for
(Eq 2.1.13)
we can write:
(Eq 2.1.14)
(
And if we take into consideration equations (Eq 2.1.12) and
(
)
for the radius we obtain:
(Eq 2.1.15)
Respectively the formula for the diameter is:
(
(Eq 2.1.16)
- Diameter of the working profile dedendum circle
Since the real profile of the teeth of the epicycloidal wheel is actually an equidistant to the theoretical
profile the following formula for the radius can be presented:
(Eq 2.1.17)
The radius of the dedendum circle is defined when we substitute equations (Eq 2.1.12) and (Eq 2.1.3) in
(Eq 2.1.17)
(
(Eq 2.1.18)
From where for the diameter of the dedendum circle will be:
(
(Eq 2.1.19)
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- Diameter of the working addendum circle
Similar to the working dedendum, the radius of the addendum is defined as difference of
radii i.e.:
and
(Eq 2.1.20)
When we substitute equations (Eq 2.1.14) and (Eq 2.1.3) for the radius and diameter respectively we
have
(
(Eq 2.1.21)
(
(Eq 2.1.22)
- Teeth depth
The depth of the teeth can be determined as the difference between the radii of the addendum and
dedendum circles. But from Figure 1 in Appendix 2 is observed that:
(Eq 2.1.23)
And after substituting the equation for
(
and (Eq 2.1.23) for
we receive the following:
(
(Eq 2.1.24)
- Coefficient of shortening of the epicycloids
The coefficient of shortening the epicycloid is designated by λ and is defined with the help of the
following relation :
(Eq 2.1.25)
Also
(Eq 2.1.26)
- Center distance
Center distance a is the distance ̅̅̅̅̅̅̅. According to Figure 1 in Appendix 2:
(Eq 2.1.27)
If we take into consideration equations (Eq 2.1.10) and that
we obtain the following equation:
(
for the pitch center distance
(Eq 2.1.28)
The center distance is not dependable on the displacement of the profile cutting instrument and is
permanent for epicycloidal wheel with a certain module and number of teeth.
16
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
-
Influence of the modification on the geometry of the teeth
The analyses of the formulas lead to the following conclusions:
a) The modification of the profile cutting instrument does not change the diameter of the pitch circle of
the epicycloidal wheel;
b) Diameter of the addendum circle
decreases when the coefficient of the modification - x of the
profile cutting instrument is increased;
c) The diameter of the dedendum circle
increases when the coefficient of the modification is
increased;
d) When the coefficient of the modification of the profile cutting instrument is increased the tooth
depth of the epicycloidal gear decreases;
e) The addendum and the dedendum circles coincide and the tooth depth is equal to zero when the
coefficient of the displacement is equal to 1.
The mention above is illustrated in Figure 12 below.
Figure 12. Lobes Modification (Alipiev, 1988)
2.1.5.Equations for teeth profile and curvature radius.
Symbol
Explanation
Coordinates of the different points
Coordinates of the different points
Angle of rotation of the generating wheel
Radius of curvature
Radius of the theoretical curvature
Coefficient of the radius of the theoretical curvature
Radius of the working curve
Radius of the position of the inflection point
Radius of the position of the inflection point
Radius of the addendum of the pin wheel
Dimension
deg
mm
mm
mm
mm
mm
mm
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Table 6.Used symbols
Note
- Determination of the theoretical and working profile of the epicycloid
The teeth profile can be determined by formation of a profile generating circle q, which will move in
relative motion. In order to do that, it is needed to form two movable Cartesian coordinate systems
(shown in Figure13) :
is the first one and it is in connection with profile generating circle ,
is the second one which is connected with the epicycloid circle. The equation of the profile generating
curve in
coordinate system is:
(
(Eq 2.1.29)
Where ξ and η are the coordinates of profile generating circle points.
Figure 13. Theoretical and Working Profile (Alipiev, 1988)
The centrodes of the profile generating curve are: r1 for epicycloid circle and rwo for generating circle.
When the generating circle turns on an angle (ψ), the epicycloid circle will turn on an angle (φ). These
angles are in relation:
(Eq 2.1.30)
The relation between
and
coordinate system is:
There is a connection between φ, ψ and
(
(
(Eq 2.1.31)
(
(
(Eq 2.1.32)
:
(Eq 2.1.33)
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Before finding the working profile p-p, first the theoretical profile T-T must be found. In this case the
radius of the generating circle will be equal to 0 (rc=0), which means that:
(Eq 2.1.34)
(Eq 2.1.35)
Using the information from above the equations for theoretical profile will be:
[(
(
[(
]
(
(
(Eq 2.1.36)
]
(
(Eq 2.1.37)
These equations for shortened epicycloid are valid in case of:
The working profile is an equidistant curve of the theoretical profile. The equations for equidistant curve
in respect of the theoretical profile are:
(Eq 2.1.32)
√
(Eq 2.1.33)
√
Where XW and YW are the coordinates of working profile; XT and YT are the coordinates of theoretical
profile.
After differentiation of XT and YT – the equations for the working profile can be written:
-
[(
(
(
[(
(
(
[(
(
(
√
[(
√
]
(
(
(
]
(
]
(Eq 2.1.34)
]
(Eq 2.1.35)
Curvature radius
The differentiation geometry shows that every curve radius can be built by the fallowing equation:
( ̇
̇ ̈
Where ̇ ̇
̈
̇ )
̈ ̇
̈ are in function of φ.
19
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq 2.1.36)
Like the teeth profile before finding the working curvature radius, it is needed to find the theoretical
( ). And after transforming and calculating the equations of theoretical teeth
curvature radius (
profile and the above equation, the equation for theoretical curvature radius will be:
(
[
(
(
(
(
Where it is obvious that
is depending only on
The connection between the working curvature radius
]
(
(Eq 2.1.37)
(
.
and the theoretical curvature radius
(
Where
is :
(Eq 2.1.38)
.
- Distinctive points of teeth profile
The teeth profile is formed by two sections: bulge section (Figure 14 (a)) and concave section (Figure 14
(b)). There are four distinctive points, which are important for the working profile (see Figure 15):
(a)
Figure 14.Profile section (Alipiev,1988)
(b)

Start point – f – is located on the dedendum circle and it has a diameter df (respectively radius rf)

Addendum point – a – is located on the addendum circle and it has a diameter da (respectively radius
ra)

Inflex point – l – at this point the teeth profile transforms from concave section to bulge section.

Border point –n – is the point where the profile has least radius. It is located on bulge section.
20
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
The exact location of l and n points can be found by the equations for their radiuses in function of φ.
√
(Eq 2.1.39)
Figure 15.Distinctive points of the profile (Alipiev, 1988)
After transforming and calculating the equations of working teeth profile and the above equation, the l
point radius and n point radius will be:
[(
[(
(
(
(
(
(
√
(
(
(
(
(
(
]
(Eq 2.1.40)
(
√
(
(
(
]
(Eq 2.1.41)
With these equations it is easy to find the radiuses of l and n points, without calculating the coordinates
of them. It is obvious that the radiuses are only depending on
.
There is one limitation - the profile generating circle rc should be less or equal than the curvature radius
.
(
√
(
(
In this case the module m will not influence over the result.
21
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq 2.1.42)
2.1.6 Forming of the epicycloidal gear transmission
Symbol
Explanation
Angle of motion transmission
Coefficient of non-centrodidity
Correction coefficient of
Dimension
deg
-
Table 7.Used symbols
Note
- Gearing conditions
Every epicycloidal gear meshes correctly if:
1. The radius of the circle on which the centers of the pins are located is equal to the pitch center
distance;
2. The radius of the pins is equal to the form generating circle;
3. The difference between the number of teeth of the epicycloidal wheel and the number of pins
must be equal to 1;
4. The center distance
is equal to the eccentricity i.e.
(
(Eq 2.1.43)
On Figure 16 is shown an epicycloidal gearing, where
and
are respectively the radii of the
centrode circles of the epicycloidal and the circle on which the pins are located. These circles are
starting circles.
Figure 16. Corrected epicycloid transmission (Alipiev, 1988)
Since :
,
22
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq 2.1.44)
for the radii of the base circles we have :
(Eq 2.1.45)
(Eq 2.1.46)
After taking into consideration the equation (Eq 2.1.43) we can determine the formulas for the radii of
the base circles
(
(Eq 2.1.47)
(
(Eq 2.1.48)
- Dimensions of the housing
All these dimension can be determined only with the basic parameters m, , x, .
Diameter of pitch circle .It is the radius on which all the pins are located. Since
and
for the radius of the pitch circle we can write down
(
If we compare the radii of the pitch and initial circles of the epicycloid plate when
obvious that
. They are equal when
.
and
(Eq 2.1.49)
it’s
Diameter of the pins. The radius and the diameter of the pins can be calculated with the following
formulas:
(Eq 2.1.50)
Diameter of the addendum circle. The radius
diameter of the cyclo gear i.e.
is dependent on the radius of the pins and the pitch
(Eq 2.1.51)
After taking into consideration of equations (Eq 2.1.49) and (Eq 2.1.50), the radius and the diameter of
addendum circle are:
(
(Eq 2.1.52)
(
(Eq 2.1.53)
23
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Diameter of the working profile dedendum circle. This diameter is calculated only for gears which are
like the one in Figure 17. The peaks of the teeth of the epicycloidal wheel should be in contact only with
the pins and that can be achieved if the following inequality
(Eq 2.1.54)
For the radius of the housing circle
to be taken into consideration
except inequality (Eq 2.1.54) there is another one which also has
(Eq 2.1.55)
(
(Eq 2.1.56)
If those inequalities are not fulfilled then the attachment of the pins to the holes in the housing is not
possible, because they will have the possibility to move radial. After applying equations (Eq 2.1.54) and
(Eq 2.1.55) and expressing the basic parameters for the diameter of the housing circle we obtain :
(
(Eq 2.1.57)
Coefficient of non-concentricity. From Figure 17 can be observed that the pins and the lobes of the
epicycloidal gear are located out of their basic circles. That increase of their size is called coefficient of
non-concentricity - l, it is defined as term of the following:
(Eq 2.1.58)
After substitution of
and
from equations (Eq 2.1.49) and (Eq 2.1.48) we receive :
(Eq 2.1.59)
2.1.7. Action line. Contact Points
The epicycloid wheel is in contact with all of the pins all the time, but only half transmit the load. The
forces F2, F3, F4 and F5 can be seen on Figure 17. Every force direction is formed by creating a line from
point P to every pin centre. These forces are normal to every contact line (K2, K3, K4 and K5). If we
assume that the epicycloid wheel is rotating anticlockwise with angular speed of ω1, it is obvious that
the transmission pins would be pins number 2, 3, 4, and 5.
24
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Figure17. Transmission Forces (Alipiev, 1988)
The meshing circle is shown on Figure 18. Point A is the beginning point of the meshing circle and point
B is the ending one. Also in the figure is shown that if housing with pins rotates with angle ψ, and that
the epicycloid wheel is rotating with an angle
. The point K is located on the line ̅̅̅̅̅ and
radius rc.
The exact location of point K can be found by looking at the triangle PCK and the following equation
can be expressed:
̅̅̅̅
(̅̅̅̅̅
̅̅̅̅
(̅̅̅̅̅
25
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq 2.1.60)
(Eq 2.1.61)
Figure 18. Meshing Circle (Alipiev, 1988)
The meshing circle equation can be found by calculating the relations between the triangles PO2Oc,
̅̅̅̅ :
DO2Oc and the lines ̅̅̅̅̅ ̅̅̅̅̅
[
[ (
√
(
(
√
]
(
(
(
(Eq 2.1.62)
]
(Eq 2.1.63)
The coordinates for each contact point from the action line can be obtained by solving (Eq 2.1.62), (Eq
2.1.63) and substituting the angle from 0 to .
2.1.8 Distinctive Meshing Angles
Symbol
Explanation
Dimension
deg
deg
Pressure angle
Angle equal to γ
Table 8. Used Symbols
Note
- Angle of Meshing
The angle of meshing
is shown in Figure 19. It has a variable quantity, depending on the contact
point in respect of the angle ψ. The value of
can be found by the following equation:
[
√
(
(
]
26
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq 2.1.64)
This angle depends on the angle ψ and on the coefficient of displacement x.
In Figure 19 it has been build the relation of
in respect of ψ with different coefficient of
displacement x. It is obvious that its maximum value is 90° and it can be as positive as negative.
in Respect of ψ (Alipiev, 1988)
Figure 19.The Angle
- Transmitting Angles
The transmitting angle is between the directors of the force ⃗⃗⃗⃗⃗ and the speed ⃗⃗⃗⃗⃗ (see Figure 20). It is
clear that this angle depends on ψ. The transmitting angle is important because of possible jamming and
it must fulfil the following condition:
Sometimes it is better to use the angle γ instead of θ. The angle γ is called angle of motion transfer and it
is the supplementary of to
.And in this case the equation will be:
Where the angle γ can be found by looking at triangle
̅̅̅̅̅̅
and using the sine theorem:
:
[
(
√
(
(
]
27
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq 2.1.65)
Figure 20.Transmitting Angles (Alipiev, 1988)
In Figure 21 it is shown the change of γ according different stages of (
(
. When
the
angle γ starts to be less than 30°, which value is not appropriate for transmitting (Alipiev,1988). The
maximum value of x can be found by looking at
again:
(
(Eq 2.1.66)
Where the angle γ is in respect of δ (with values from 0 to π). After looking close to this equation , the following can be expressed:
(Eq 2.1.67)
And after calculating the above equations – the maximum value for γ is:
(
Figure21. The Change of γ According Different Stages of x (Alipiev, 1988)
28
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq 2.1.68)
2.1.9 Contact ratio
Symbol
Explanation
Overlap ration angle
angle
angle
Contact ratio angle
allowable angle of transmitting
Angle step
Contact ratio
Optimal contact ratio
Dimension
deg
deg
deg
deg
deg
deg
-
Table 9. Used Symbols
Note
- Angle of overlap ratio
The angle of beneficial overlapping
is actually the angle of rotation of the gear, to which the contact
between the teeth is done with transmitting angles higher than
. In initial position
the angle of transmition is
. When the epicycloid gear starts to rotate the angle increases and
when
the angle becomes equal to the allowable angle of transmitting (
). If the rotating
continues the angle increases but to a certain point where
(
. After that it starts to
decrease and when
the angle is equal to . The decrease will continue until
in that
condition
. The formula for the angle of beneficial overlapping is :
(Eq 2.1.69)
Since some mistakes can occur while making the profile, there is another more accurate way to
determine the angle. If we substitute
and
in (Eq 2.8.2) we have :
(
(
in which with
(
(
(Eq 2.1.70)
is substituted .
The answers of the biquadratic equation - (Eq 2.1.70) are:
√(
(Eq 2.1.71)
√(
(Eq 2.1.72)
After substituting (Eq 2.1.71) and (Eq 2.1.72) in equation (Eq 2.1.69) for the angle of beneficial
overlapping can be written :
[
√(
]
[
√(
29
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
]
(Eq 2.1.73)
Or after simplifying :
(
)
(Eq 2.1.74)
- Overlap ratio
The overlap ratio is related to the continual and smooth meshing of the teeth profiles. In order to have
smooth meshing it is necessary to have as much as possible meshed pairs of teeth.
In epicycloidal gears half of the teeth in any moment are in contact with the pins, meaning that the
coefficient of the overlapping should be equal to half of the pins. Since they are involved in transmitting
the load. The overlap ratio is defined as a relation of the angle of overlapping
to the angular
pitch of the housing pins
. It can be expressed:
(Eq 2.1.75)
If the coefficient is defined like equation (Eq 2.1.75) it is not clear enough how many teeth pairs are
engaged. Since that, a new term is introduced beneficial overlap ratio:
(Eq 2.1.76)
Where,
can be determined by equation (Eq 2.1.74).
If we take into consideration equations (Eq 2.1.74) and (Eq 2.1.76) for the coefficient of beneficial
overlap ration can be written down the following:
(
)
(Eq 2.1.77)
The coefficient is dependable on the number of pins , modification and the allowable angle of
transmitting . If the number of pins
increases then
increases too, but if x increases then
decreases.
- Boundary conditions of the modification
In the process of designing of corrected epicycloidal gears it is essential to take into consideration the
modification of the profile cutting instrument x. The real range of variation of x is between
and
(
). The minimal values for can be determined with the following equation:
√
(
(
(Eq 2.1.78)
If x is increased that leads to decreasing the beneficial overlap ratio. To be sure that the transmission is
working properly it is necessary the following to be fulfilled:
,
30
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq 2.1.79)
in which with
is designated the allowable minimal value of the beneficial overlap ratio.
According to equation (Eq 2.1.77) for the maximum value of the coefficient of modification of the
profile cutting instrument can be written :
(
)
(Eq 2.1.80)
2.2. Cycloid gearbox design
2.2.1 Forces and Force distribution in the cycloid gear
The cycloid reducer as a mechanical system is a statically indeterminable system. Therefore to obtain
the stresses it is required to build a dynamic model in an appropriate software environment. The
computer-aided approach will give very high accuracy.
Research made in National Academy of Sciences of Belarus (Tsetserukou D., 2012), presents force
distribution among the transmission elements. In Figure 22 is shown how exactly the load is distributed
among the housing pins. There can be seen that half of the pins are always in contact with the gear
(colored with light blue) but the most loaded are less than half (colored with dark blue). These types of
transmissions allow an overload of five times their rated torque. This is due to the fact that all the time
while transmitting a large contact surface is provided.
Figure 22. Force distribution between the gear and the pins (Tsetserukou D., BasinukV. ,2012)
Looking in depth a couple of forces will be acting on the main elements. Shown in Figure 23, below is a
scheme how they are acting and their directions in a Cartesian coordinate system.
31
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Figure 23. Forces (Yao & Zhang , 1997)
The acting forces here are:
-Force P is acting between the cycloid disc and the pins. Its value can be determined according to
the following equation:
√(
With
(Eq. 2.2.1)
is designated the force acting on axis x:
(Eq. 2.2.2)
Where,
is the number of pins in the housing,
is the input torque,
- coefficient of width
shortening of the epicycloid wheel,
- number of lobes and - pitch radius of pins.
The force
is on the Y axis:
∑
[
(
)
]
[
(
(
)
)]
(Eq.2.2.3)
Force F is acting between the eccentric bearing and the cycloid disc (driving force) and its value can be
determined by:
(
(
)
(
(
(Eq.2.2.4)
)
32
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq.2.2.5)
And the total force can be determined:
√(
(Eq.2.2.6)
Force is acting between the output shaft rollers and disc. Its value can be determined with equation
(2.2.7).
2.2.2. Designing the holes and the thickness of the cycloid discs
-Diameter of output pins . In Figure 24 is shown where these diameters are situated.
Figure 24. Corrected Cycloid Drive (Rao Zhengang, 1994)
Where:
is the output pins’ diameter,
is the roller diameter and with
is designated the disc’s
hole diameter. To calculate the diameter, first the force - (acting on the rollers shown on figure below)
should be taken into consideration. The force is calculated by the following equation:
Tg
Zw
(Eq2.2.7)
Where with is designated the torque per cycloid disc. The number of holes in the gear for the output
pins is presented with
; and is referred to the radius on which the pins are located.
According to (Zhengang, R.,1994) the output torque (
epicycloid discs (N) , e.g.:
is evenly distributed between the number of
(Eq 2.2.8)
33
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
On Figure 25 is shown a section cut of the reducer where the force is shown with a blue arrow.
Figure 25. Reducer Section Cut (Zhengang,1994)
The maximum bending stress in a pin is:
(Eq 2.2.9)
Where with W is designated the section module:
(Eq 2.2.10)
The distance L is the lever of the bending moment created from the force
calculated using this equation:
L
B
δ
on the pin. It will be
(Eq 2.2.11)
Where, B is the thickness of the disc and δ the distance between the discs. Also to calculate the pins for
the output shaft the bending stress ( P is needed. It can be found by the following equation:
P
With
b
b
(Eq 2.2.12)
here is designated the tensile stress.
The equation for the diameter of the output pins:
Tg ( 5B δ
√Z
w Rw σFP
34
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq 2.2.13)
When the diameter
[
[
]
]
When
12
17
is known,
14
20
can be easily found according to the Table 10:
17
26
is known the hole on the gear (
Table 10. Bush outer diameter (Rao, 1994)
26
32
35
45
55
38
45
50
60
75
22
32
) can be calculated using this equation:
(Eq 2.2.14)
The modern reducers have different output shaft pins. The shape is based on the area provided by the
calculated diameter . The better shape provides more contact surface and eliminates the roller element.
Reducing the production cost per unit.
Figure 26.Pin Shape (Profile)
On Figure 26 is shown the shape, it is based on the inscribed in the trapezoid circle. This way the needed
surface is provided. The edges are rounded and when the pins are offset inward with the value of the
eccentricity, they can fluctuate in both of the discs. This provides a steady rotation of the output shaft.
-
Disc thickness.
It is very important to determine the disc thickness with respect to the material and the contact forces.
Here is shown a simplified equation, giving satisfactory result, but in the authors opinion more
investigation in the area is needed for getting optimal designs.
35
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Figure 27 shows section cut of the cycloid discs pins. With B is marked the thickness of the cycloid
discs.
Figure 27. Section Cut (Rao , 1994)
The thickness can be calculated by the following equation:
(
Where
is pitch radius of the housing.
36
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(Eq 2.2.15)
3. Method
This chapter describes briefly and clearly the way of performing the work. Here will be explained how
the computer software is used and the steps taken to receive the results.
3.1. Getting the results
- Examine the literature and theory
In order to find results and answers, first of all the literature and the theory in chapter 2 should be
examined in details. The way of creating all the geometry and the relations should be clearly understood.
Further knowledge in the engineering terminology, practical geometry and advanced computer skills in
3D modelling and simulations are also necessary due to the complexity of the material.
Before creating the 3D models and assembly, analytic calculations are needed. At the beginning the part
geometry needs to be generated, using equations in chapter 2.Below is explained how this has been
performed.
- Modeling in SolidWorks
SolidWorks software is a CAD (computer aided design) environment which gives the opportunity to
create 3D solid body models. The whole process of creating a assembled product includes two steps:
parts and making assemblies. The first and actually the basic step, creates parts (solid bodies) from
already existing sketches, using ‘features’. The second step allows mounting the solid parts in particular
order with the relations needed for the proper working of the mechanism. Using the Drawing module
technical drawings from the parts or assemblies are created.
Using the potential that SolidWorks possess a prototype\model has been made and tested. Creating the
working profile of the cycloidal gear and successfully assembling it with the other components. A
special sketch feature called “equation driven curve” is used for establishing in the parametric equation
for the needed curve or in this case profile. As a result a perfect epicycloidal curve is obtained. Using the
feature extrude boss/base the epicycloidal wheel is created. For the simulation a feature called Motion
Study is used, in which rotational speed is added on a certain axis.
- Parts creating
All of the components are built in Part files main data type file of SolidWorks. The output file format is
*.SLDPRT. The most important step is making the epicycloid wheel (disc). Since SolidWorks does not
have tools and commands for making epicycloid profiles, this specific profile has been built by
“equation driven curve” command; the outcome is displayed in figure 28 (a) and (b). This command
requires two parametric equations –for X and Y, which are functions of t. The equations for the working
profile are from the theory chapter 2. They have been simplified and modified, in order to be understood
by the software, as follows:
((
(
(
((
(
(
(
(
(
(
(
(
((
(
(
(
(
(
(
(
((
(
Where the value of tis set to be from 0 to 2*pi.
37
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
After building the 2D sketch with the required profile, “Extruded Boss/Base” command has been used to
build in 3D (Figure 28 (b)). The command “Extruded Cut” is used for making the holes in the disc.
Next step includes the housing. It has been built by the ordinary SolidWorks tools and commands, like:
“sketch circle”, “sketch arc”, “Extruded Boss/Base”, ” Extruded Cut” etc.
(a).Sketch Profile
(b). Extruding the Profile
Figure 28
3.2. Making Assembly and Simulation
After all of the parts have been designed, they have been assembled. It can be made in Assembly module
of SolidWorks. The output file format is *.SLDASM.
After importing the parts, only mates are added. Using the “Mate” command. We have made the entire
assembly by the standard mates, available in SolidWorks: “Coincident”, “Parallel”, “Perpendicular”,
“Tangent”, “Concentric”, etc ,except the case where the gears are mated using “Mechanical Mate”.
SolidWorks has a simulation package to set up virtual environments so that a test on a product can be
tested before manufacture. Test against a broad range of parameters — like durability, static and
dynamic analyses, motion of assemblies, heat transfer, fluid dynamics, etc.
To make the tests four steps have been accomplished:
1.
2.
3.
4.
5.
The geometry has been defined
Material has been assigned
The Study has been specified
Environment has been set
The System processed the calculations and presented results
38
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
This modeling system offers a Von Mises – linear static analysis. Where the following assumptions are
made by the software:
-The involved material is linear, when reacting to a force the displacement is proportional to the applied
force, like the spring in figure 29 below (e.g. the material obeys Hooks law);
-Deformations are relative to the tested component or assembly’s size;
-The loads do not change in time (e.g. the loads are static).
Figure 29. Force acting on a spring
(http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Dynamics/gifs/Forces04.gif)
The SolidWorks static analysis tool works with Von Mises yield criterion. According to this criterion the
yield begins when the strees (known as the yield strength) reaches a critical value. Also it is based on the
octahedral shear stress. The Von Mises stress criterion is used to predict yielding under any loading
condition from results of simple uniaxial tensile tests (Asaro R, 2004, page 463).
39
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
4. Results
This chapter presents the output data of this thesis. The results are reported simply and clearly. Most of
the results are obtained by computer simulations with advanced software computations. In order to be
understandable the results are simplified.
The results from the theory part are a new way of looking at the geometric shapes of the reducer. We
have used Professor Ognian Alipiev’s unique approach to design a product together with the group,
working on the shafts, bearings etc. In order to fulfil the demands, we have designed the cycloid gear
using standard values for the module, used in spur gears. As explained this is not typical when working
with such transmissions. In this case this approach can be standardized in future for these kinds of gears.
They can help the people working with cycloid transmissions.
All of the results are presented after examination and analysing the equations and the information in the
theory chapter. The design process has been achieved from the output data given by SwePart
Transmissions AB. What they required is the torque on the output shaft of the reducer should be 6300
Nm, ratio 240 and rotational speed of 4,3 rpm. To obtain these results, the forces and force distribution
have been found.
4.1. Total ratio
The total ratio calculated with equation 1.1 in the background is as follows:
(1.1)
4.2. Solidworks Simulation Results
Because of the complexity of the profile of the cycloid wheel we used computer simulation to get
appropriate results. Also it is necessary to use the help of the computer software because the limitation
of time, which is not enough for analytic calculations.
For this scheme the model of the gearbox has been simplified significantly, removing all the components
except the housing, cycloid gear and eccentric shafts. Also the output, input and eccentric shaft are
simulated separately without being connected with other parts of the cycloid reducer. These
simplifications do not affect significantly the results. These assumptions and simplifications have been
done to reduce the calculation time as such analyses take a lot of time if not simplified. The results are
presented as figures, tables and diagrams for the model information, stress, displacement, strain and
safety factor of the elements. Below are simulated the cycloid gear stage (cycloid disc and housing with
pins), eccentric, output and input shafts.
4.2.1. Cycloid Gear Results – The results are presented in Appendix 1C.
4.2.2. Input Shaft Results - The results are presented in Appendix 2C.
4.2.3. Output Shaft Results - The results are presented in Appendix 3C.
4.2.4. Eccentric Shaft Results - The results are presented in Appendix 4C.
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
5. Analysis
Chapter analysis introduces principles, relationships or generalisations from the results. Sometimes the
results speak for themselves. All of the results can be found in chapter 4. The function of the analysis
chapter is to describe the ideas, models and theories with comparison of these with the computational
data.
The Cycloid speed reducer has been studied, in details. Sufficient calculation process for the geometry
of the cycloid gear has been presented. Using hi-end, up to date CAE software, the calculated model was
studied showing the many benefits of the mechanism. In the future the need for such machines will
continue to rise, as the reducer can be implemented almost in every mechanized solution.
Chapter 4.1 shows the total ratio (i) result. It has been calculated to be 240,47. According to the given
input data in chapter 1.3 it should be 240. Comparing the two values - it is obvious that the designed
reducer ratio is in theory as close as possible to the desired one. The result is based on analytic
calculation, which makes it accurate and precise.
In the previous chapter are shown the graphical tables containing the outcome from the CAE software.
For every tested model there are presented the following table types:
-
Model info – shows general information about the tested model
Study properties – they show information about the study conditions
Units – the used units in the study
Materials – there is displayed the type of material and its physical properties
Loads and fixtures – information about the loads and the used fixtures
Contact information – about how the software defines the contact conditions
Mesh information – there can be observed a meshed view and detail about the meshing
Stress – these are diagrams where with a color-graphic method is displayed the resultant stresses
Displacement – it can be observed the linear displacements also in color-graphic
Strain – strain level in the tested part
Factor of safety – there with red color are designated the zones where safety factor is below 2 (e.g.
)
Here will be analyzed the stress/displacement/strain/factor of safety diagrams.
- Cycloid gear and housing
The model information given in Table 4.1 shows included parts (gear and housing) and the basic
information about them like mass, volume, density and weight. Also with purple color can be seen the
torque vectors that has been set. All the details (temperature, analysis types etc.) about the simulation
and study properties are shown in Table 4.2. All of these simulations use International System Units (SI)
(Table 4.3). This means that the pressure is in MPa, the temperature is in Kelvin etc.
The material properties are included in Table 4.4, where it is shown the yield and tensile strength of the
used material. SolidWorks does not have the exact material as in our case (20 MnCr 6), this is the reason
why we have set the material to be alloy steel (SS). It has lower yield and tensile strengths, comparison
to 20 MnCr 6. This means that if the simulation is successful with this material, the design of the cycloid
wheel would be good enough for material with higher values of yield and tensile stresses.
The applied torque in cycloid wheel is 3200Nm (Table 4.5).The simulation is set to be with global
contact (Table 4.6), which means that contact will occur between the cycloid wheel and the housing.
Also the mesh quality has been set to be high (Table 4.7), giving more accurate results.
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Looking at the Table 4.8 can be observed that the developed stresses in the mechanical system do not
exceed the yield strength of the material (640,4MPa). On the color scale at right, can be seen the
meaning of every color occurred in details. With blue color and nuances of blue are colored the areas
where the stresses have a low values (from 0 to ≈200MPa) of the stresses. Respectively with red color
the high values of the stresses. Fortunately when looking at the results for the cycloid gear design, it can
be realized that overall the cycloid gear-housing has withstood the loads. There are not places with
values more than ≈250MPa (only blue and some nuances of blue color). This means that all of the stress
values are times lower than the yield strength of the material. In this case we are sure that the cycloid
wheel and housing are designed properly, because in reality the gears and the pins will probably be
tempered or/and galvanized.
In Table 4.9 is shown the displacement of the cycloid wheel. Analogically as stress analysis the color
scale at right shows the value of the displacement (in mm). The minimum and maximum values are
respectively 0 and 27,66µm. Most of the displacements in the cycloid wheel are between 10µm and
20µm, except one area with values more than 20µm (indicated with yellow color). These values are still
under the nominal values in which case no problems will occur.
With Table 4.10 is shown the permanent strains after unloading the components. As can be read from
the diagram, all strains are in elastic zone (low values are colored in blue).
The safety factor is shown in Table 4.11. There is not color scale in this diagram. The reason is that only
two colors exist in this type of analysis. With blue are colored the areas with safety factor bigger than
two, and with red are colored the areas with safety factor less than two (red < SF=2 < blue). As easily
can be seen in the figure – all zones of the cycloid wheel and the housing have safety factor bigger than
two (zones with blue color).
These static analyses give a satisfactory accuracy as the conditions simulated by the software are very
similar to the real working condition.
- Input Shaft
Analogically this analysis provides the same information in tables from 4.12 to 4.17 as tables from 4.1 to
4.7 (as in the previous analysis). The most significant deference is in tables from 4.18 to 4.21.
In table 4.18 can be observed that the developed stresses in the mechanical system do not exceed the
yield strength of the material (220,6MPa). On the color scale at right, can be seen the meaning of every
color occurred in the details. With blue color and nuances of blue are colored the areas where the
stresses have a low values (from 0 to ≈100MPa) of the stresses. Respectively with red color are colored
the high values of the stresses. Fortunately when looking at the results for the input shaft, it can be
realized that it has withstood the loads. There are no places where the values are more than ≈115MPa
(only blue and some nuances of blue color). This means that all of the stress values are times lower than
the yield strength of the material. In this case we are sure that the input shaft is designed properly.
In Table 4.19 is shown the displacement of the input shaft. Analogically as stress analysis the color scale
at right shows the value of the displacement (in mm). The minimum and maximum values are
respectively 0 and 45,62µm.
With Table 4.20 is shown the permanent strains after unloading the components. As can be read from
the diagram, all strains are in elastic zone (low values are colored in blue and green).
The safety factor is shown in Table 4.21. There is not color scale in this diagram. The reason is that only
two colors exist in this type of analysis. With blue are colored the areas with safety factor bigger than
two, and with red are colored the areas with safety factor less than two (red < SF=2 < blue). As easily
can be seen in the figure – all zones of the input shaft have safety factor bigger than two (zones with
blue color).
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
- Output Shaft
In table 4.22 is displayed the model information, such as volumetric properties, view etc. In the next
table 4.23 are the details about the properties of the study and the used features by the software. The
next two tables 4.24 and 4.25 prove units and material information.
In Table 4.26 are shown the used fixtures when doing the simulation and also the loads. Here one face is
fixed, there is one that is where the bearings will be and its status is made Roller/Slider to simulate the
bearing effects. The opposite to the fixed face is the loaded face subjected to a torque of 6400Nm, the
actual output torque.
Looking down to the next Table 4.27. It includes a lot of data about the mesh and mesh control. There is
also a picture showing the meshed detail, there can be observed that the meshing is quite fine, which will
help generate a very accurate calculation process.
On the stress color-diagram in Table 4.28, can be seen the resultant stresses from the simulated twisting.
The deformed state of the model is magnified by the software, only to help the viewer to imagine how
this will happen in reality. Again here can be seen that the stresses do not exceed the yield strength of
the material as the colors show. Also the next three figures (4.29,4.30,4.31) show that overall the design
will withstand the static loads.
- Eccentric Shaft
The following tables have the similar information, they provide information how the study was
conducted and etc., from Table 4.32 up to 4.37.
In Table 4.38 is shown the stress color-diagram. Looking at the colors and the color legend, it can be
said that the stress does not exceed the material yield strength. Also the same can be said for the other
color-diagrams presenting the displacement, strain and factor of safety.
43
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
6. Conclusion
This study shows the geometry and designing and calculating process in a second stage of a cycloidal
reducer. The cycloidal gear capacity can be estimated by computer aided software like SolidWorks, as
shown in this work. All of the theory and the equations are based on advanced work of many high
qualified scientists in this area of study. By this thesis we have created SolidWorks assembly models,
which can be used for further analysis by recalibrating the materials and model settings. These models
have been simulated in software environment, in order to get results.
The results included in chapter 4 are accurate enough for making predictions and important conclusions.
The designing process in Appendix 1 and theory chapter is shown in clear way. By following the
equations it is easy to find all dimensions and designing the cycloid gear components.
As the results and analysis present, this type of gear reducers are more reliable and profitable than the
conventional spur gears reducers. More researches and examinations are required in this area. Our team
hopes that this work will help for future studies and standardization of the cycloid reducer.
44
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
7. References

Alipiev O. (1988), “Geometry and Forming of Epi- and Hypo-Cycloidal Toothed Wheels in
Modified Cyclo-Transmission”, Ph.D. Dissertation, Ruse, pp.1-36 (in Bulgarian).

Ashby M. (2005), How to Write a Paper 6rd Edition, Engineering Department, University of
Cambridge, Cambridge

Darali drives, (2012) homepage, http://www.darali.com/page17.html

D’Amico J., (2012), http://machinedesign.com/article/comparing-cycloidal-and-planetarygearboxes-0203

Sumitomo Drives, (2012) homepage,
http://www.smcyclo.com/modules.php?name=Product&op=productBrand&brand_id=7

TsetserukouD., V. Basinuk (2012), CONTACT FORCE DISTRIBUTION AMONG PINS OF
TROCHOID TRANSMISSIONS, Department of vibroprotection of machines, Institute of
Mechanics and Reliability of Machines of the National Academy of Sciences of Belarus (IMRM
of NAS of Belarus), Akademicheskaya 12, 220072 Minsk, Belarus

Yao W.,Zhang Z. (1997), Journal of Beijing, Institute of Machinery Industry(In Chinese)

Rao Z., (1994),”Planetary transmission mechanism design”(in Chinese)

Robert J. (2004), V. Lubarda , Mechanics of Solids and Materials, University of Cambridge,
Cambrige

http://www.nabtescomotioncontrol.com/pdfs/RVseries.pdf

http://machinedesign.com/article/comparing-cycloidal-and-planetary-gearboxes0203?page=0%2C0

http://fluid.ippt.gov.pl/ictam04/CD_ICTAM04/SM2/10540/SM2_10540_new.pdf

http://www.rmhoffman.com/faqs/engineering/cycloidal.html
45
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
8. Appendix
Appendix 1 – Geometry Design
Appendix 2 – Figures
Appendix 3 – Linear Static Analyses with SolidWorks
Appendix 4 - Drawings
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Appendix 1 - Geometry Design
1A. Basic Parameters
2A. Epycycloidal Wheel Dimensions
3A. Pin Wheel Dimensions
4A. Epicycloidal Gear Parameters
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Appendix 1- Geometry Design
1A. Basic Parameters
The calculation process starts with determination of the four basic parameters of the epicycloidal wheel:
Number of teeth - according to (Eq 1) and (Eq 1.1)
Module – by standard value:
Coefficient of the radius of the profile generating circle:
(recommended by (Alipiev, 1988))
coefficient displacement of the profile cutting instrument*:
(
where
)
, what else is not known is
(
(
√
(
.
)
(
)
In the formula
is substituted with 0,49 which is the maximum coefficient of modification of the
profile cutting instrument for this kind of gears. Finally for x we have
2A. Epicycloidal Wheel Dimensions
2.1. Diameter of pitch circle:
(Eq 2.1.11)
2.2. Diameter of the theoretical dedendum circle:
(
(Eq 2.1.13)
(
2.3. Diameter of theoretical addendum circle:
(
(Eq 2.1.16)
(
2.4. Diameter of dedendum circle:
(Eq 2.1.19)
(
(
2.5. Diameter of addendum circle:
2
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
(
(Eq 2.1.22)
(
2.6. Teeth height:
(
(Eq 2.1.23)
(
2.7. Addendem curvature radius*:
[
(
(
(
]
[
]
[
(
]
[
]
(
2.8. Dedendum curvature radius*:
[
(
(
(
(
(
(
(
]
2.9. Minimal teeth curvature radius*:
[
(
√
(
(
(
√
(
(
2.11. Coefficient of shortening of the epicycloid:
(Eq 2.1.26)
1.12. Disc Thickness
(
(Eq 2.2.9)
3A. Pin Wheel Dimensions.
3.1. Number of teeth of the pin wheel:
3.2. Diameter of the pitch circle:
3.3. Diameter of the pins:
(Eq.2.1.50)
3
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
]
3.4. Diameter of the addendum circle:
(
(Eq 2.1.53)
3.6. Contact angle*
(
(
(
4A. Epicycloidal Gear Parameters.
4.1. Centre distance
(
(
4.2. Diameters of initial circles:
-epicycloidal wheel
-pins wheel
(
(
(
(
4.3. Coefficient of non centoroidity
4.4. Maximum angle of transmitting movement
(
(
4.5. Angle of beneficial overlapping
(
)
(
)
(
)
4.6. Coefficient of beneficial overlapping
(
)
* - Equations marked with * symbol are taken directly from (Alipiev, 1988), based on the theory in chapter 2.
4
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Appendix 2 - Figures
Figure 1. Gear dimensions (Alipiev, 1988)
5
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Figure. 2 Features and Benefits of the RV (Nabtesco,2012 homepage)
6
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Figure 3. Cycloid Implementation Examples (Nabtesco,2012 homepage)
7
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Appendix 3 – Linear Static Analyses with SolidWorks
1C. Cycloid Gear Results
2C. Input Shaft Results
3C. Output Shaft Results
4C. Eccentric Shaft Results
8
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
1C. Cycloid Gear Results
Table 4.1 Model Information
Model name: Cycloid Gear Model
Current Configuration: Default
Solid Bodies
Document Name and
Reference
Gear
Treated As
Volumetric Properties
Mass:5.11346 kg
Volume:0.000664086 m^3
Density:7700 kg/m^3
Weight:50.1119 N
Solid Body
Housing
Mass:11.1377 kg
Volume:0.00144645 m^3
Density:7700 kg/m^3
Weight:109.149 N
Solid Body
9
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Document Path/Date
Modified
C:\Users\DELLi7\Deskto
p\epicycloid\Part1.SLDP
RT
May 23 11:50:18 2012
C:\Users\DELLi7\Deskto
p\epicycloid\Part2.SLDP
RT
May 23 11:51:10 2012
Table 4.2 Study Properties
Study name
Analysis type
Static Study 1
Static
Mesh type
Solid Mesh
Thermal Effect:
On
Thermal option
Include temperature loads
Zero strain temperature
298 Kelvin
Include fluid pressure effects from SolidWorks Off
Flow Simulation
FFEPlus
Solver type
Inplane Effect:
Off
Soft Spring:
Off
Inertial Relief:
Off
Incompatible bonding options
Automatic
Large displacement
Off
Compute free body forces
On
Friction
Off
Use Adaptive Method:
Off
Table 4.3 Units
Unit system:
SI (MKS)
Length/Displacement
mm
Temperature
Kelvin
Angular velocity
Rad/sec
Pressure/Stress
N/m^2 (Mpa)
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Table 4.4 Material Properties
Model Reference
Properties
Components
Name: Alloy Steel
Model type: Linear Elastic
Isotropic
Default failure Unknown
criterion:
Yield strength: 6.20422e+008
N/m^2
Tensile strength: 7.23826e+008
N/m^2
Elastic modulus: 2.1e+011 N/m^2
Poisson's ratio: 0.28
Mass density: 7700 kg/m^3
Shear modulus: 7.9e+010 N/m^2
Thermal expansion 1.3e-005 /Kelvin
coefficient:
SolidBody - Gear
SolidBody - Housing
Curve Data:N/A
Table 4.5 Load
Load name
Load Image
Load Details
Entities:
Reference:
Type:
Value:
Torque-1
Table 4.6 Contact Information
Contact
Contact Image
Global Contact
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
1 face(s)
Face< 1 >
Apply torque
3200 N-m
Contact Properties
Type: Node to node
Components: 1
component(s)
Table 4.7 Mesh Information
Mesh type
Solid Mesh
Mesher Used:
Curvature based mesh
Jacobian points
4 Points
Maximum element size
13.292 mm
Minimum element size
2.65841 mm
Mesh Quality
High
Table 4.8 Stress
Name
Stress1
Type
VON: von Mises Stress
Min
0.000959013 N/mm^2
(MPa)
Node: 172898
Assem1-Study 1-Stress-Stress1
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Max
1146.83 N/mm^2
(MPa)
Node: 97468
Table 4.9 Displacement
Name
Displacement1
Type
URES: Resultant Displacement
Min
0 mm
Node: 76504
Max
0.0276609 mm
Node: 28637
Assem1-Study 1-Displacement-Displacement1
Table 4.10 Strain
Name
Strain1
Type
ESTRN: Equivalent Strain
Min
1.21659e-009
Element: 105555
Assem1-Study 1-Strain-Strain1
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Max
0.00311473
Element: 35303
Table 4.11 Factor of Safety
Name
Type
Factor of Safety1
Max von Mises Stress
Min
0.540987
Node: 97468
Assem1-Study 1-Factor of Safety-Factor of Safety1
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Max
646938
Node: 172898
2C. Input Shaft Results
Table 4.12. Model Information
Model name: ImputShaft
Current Configuration: Default
Solid Bodies
Document Name and
Reference
Chamfer3
Treated As
Solid Body
Volumetric Properties
Mass:0.31655 kg
Volume:4.05834e-005 m^3
Density:7800 kg/m^3
Weight:3.10219 N
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Document Path/Date
Modified
C:\Users\DELLi7\Desktop\I
mputShaft.SLDPRT
May 24 18:06:48 2012
Table 4.13. Study Properties
Study name
Study 1
Analysis type
Static
Mesh type
Solid Mesh
Thermal Effect:
On
Thermal option
Include temperature loads
Zero strain temperature
298 Kelvin
Include fluid pressure effects from SolidWorks Flow
Simulation
Solver type
Off
Inplane Effect:
Off
Soft Spring:
Off
Inertial Relief:
Off
Incompatible bonding options
Automatic
Large displacement
Off
Compute free body forces
On
Friction
Off
Use Adaptive Method:
Off
Result folder
SolidWorks document (C:\Users\DELLi7\Desktop)
FFEPlus
Table 4.14. Units
Unit system:
SI (MKS)
Length/Displacement
mm
Temperature
Kelvin
Angular velocity
Rad/sec
Pressure/Stress
N/m^2
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Table 4.15. Material Properties
Model Reference
Properties
Name:
Model type:
Default failure criterion:
Yield strength:
Tensile strength:
Elastic modulus:
Poisson's ratio:
Mass density:
Shear modulus:
Thermal expansion
coefficient:
Components
Plain Carbon Steel
Linear Elastic Isotropic
Unknown
2.20594e+008 N/m^2
3.99826e+008 N/m^2
2.1e+011 N/m^2
0.28
7800 kg/m^3
7.9e+010 N/m^2
1.3e-005 /Kelvin
SolidBody
1(Chamfer3)(ImputShaft)
Curve Data:N/A
Table 4.16. Load
Load name
Load Image
Load Details
Entities:
Type:
Value:
Torque-1
Table 4.17. Mesh Information
Mesh type
Solid Mesh
Mesher Used:
Standard mesh
Automatic Transition:
Off
Include Mesh Auto Loops:
Off
Jacobian points
4 Points
Element Size
1.71883 mm
Tolerance
0.0859414 mm
Mesh Quality
High
Total Nodes
122152
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
1 face(s)
Apply torque
26.3 N-m
Total Elements
84271
Maximum Aspect Ratio
52.914
% of elements with Aspect Ratio < 3
99.5
% of elements with Aspect Ratio > 10
0.00119
% of distorted elements(Jacobian)
0
Time to complete mesh(hh;mm;ss):
00:00:07
Computer name:
DELLI78GB
Mesh Control Name
Mesh Control Image
Mesh Control Details
Entities:
Units:
Size:
Ratio:
Control-1
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
36 edge(s)
mm
0.859414
1.5
Table 4.18. Stress
Name
Type
Min
Max
Stress1
VON: von Mises Stress
0.0945698 N/mm^2
(MPa)
Node: 2916
134.885 N/mm^2
(MPa)
Node: 94615
ImputShaft-Study 1-Stress-Stress1
Table 4.19. Displacement
Name
Type
Min
Max
Displacement1
URES: Resultant Displacement
0 mm
Node: 301
0.0456185 mm
Node: 434
ImputShaft-Study 1-Displacement-Displacement1
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Table 4.20. Strain
Name
Type
Min
Max
Strain1
ESTRN: Equivalent Strain
8.12714e-007
Element: 82232
0.000325481
Element: 17374
ImputShaft-Study 1-Strain-Strain1
Table 4.21. Factor of Safety
Name
Type
Min
Max
Factor of Safety1
Max von Mises Stress
1.63542
Node: 94615
2332.61
Node: 2916
ImputShaft-Study 1-Factor of Safety-Factor of Safety1
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
3C. Output Shaft Results
Table 4.22. Model Information
Model name: Output Shaft
Current Configuration: Default
Solid Bodies
Document Name and
Reference
Chamfer3
Treated As
Volumetric Properties
Mass:26.5767 kg
Volume:0.00340727 m^3
Density:7800 kg/m^3
Weight:260.452 N
Solid Body
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Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Document Path/Date
Modified
C:\Users\DELLi7\Desktop\
Solid\Part10.SLDPRT
May 24 18:20:17 2012
Table 4.23. Study Properties
Study name
Study 1
Analysis type
Static
Mesh type
Solid Mesh
Thermal Effect:
On
Thermal option
Include temperature loads
Zero strain temperature
298 Kelvin
Include fluid pressure effects from SolidWorks
Flow Simulation
Solver type
Off
Inplane Effect:
Off
Soft Spring:
Off
Inertial Relief:
Off
Incompatible bonding options
Automatic
Large displacement
Off
Compute free body forces
On
Friction
Off
Use Adaptive Method:
Off
Result folder
SolidWorks document
(C:\Users\DELLi7\Desktop\Solid)
FFEPlus
Table 4.24. Units
Unit system:
SI (MKS)
Length/Displacement
mm
Temperature
Kelvin
Angular velocity
Rad/sec
Pressure/Stress
N/m^2
22
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Table 4.25. Material Properties
Model Reference
Properties
Name:
Model type:
Default failure criterion:
Yield strength:
Tensile strength:
Elastic modulus:
Poisson's ratio:
Mass density:
Shear modulus:
Thermal expansion
coefficient:
Components
Plain Carbon Steel
Linear Elastic Isotropic
Unknown
2.20594e+008 N/m^2
3.99826e+008 N/m^2
2.1e+011 N/m^2
0.28
7800 kg/m^3
7.9e+010 N/m^2
1.3e-005 /Kelvin
SolidBody 1(Chamfer3)(Part10)
Curve Data:N/A
Table 4.26. Loads and Fixtures
Fixture name
Fixture Image
Fixture Details
Entities:
Type:
3 face(s)
Fixed Geometry
Fixed-1
Resultant Forces
Components
Reaction force(N)
Reaction Moment(N-m)
X
13.7777
0
Y
0.888504
0
Z
-0.321193
0
Entities:
Type:
Resultant
13.8101
0
1 face(s)
Roller/Slider
Roller/Slider-1
Resultant Forces
Components
Reaction force(N)
Reaction Moment(N-m)
X
-16.1683
0
Y
124.928
0
23
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Z
15.6023
0
Resultant
126.932
0
Load name
Load Image
Load Details
Entities:
Type:
Value:
Torque-1
Table 4.27. Mesh Information
Mesh type
Solid Mesh
Mesher Used:
Standard mesh
Automatic Transition:
Off
Include Mesh Auto Loops:
Off
Jacobian points
4 Points
Element Size
7.52525 mm
Tolerance
0.376263 mm
Mesh Quality
High
Total Nodes
111439
Total Elements
73128
Maximum Aspect Ratio
9.584
% of elements with Aspect Ratio < 3
97.5
% of elements with Aspect Ratio > 10
0
% of distorted elements(Jacobian)
0
Time to complete mesh(hh;mm;ss):
00:00:08
Computer name:
DELLI78GB
24
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
1 face(s)
Apply torque
6400 N-m
Mesh Control Name
Mesh Control Image
Mesh Control Details
Entities:
Units:
Size:
Ratio:
Control-1
25
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
73 edge(s)
mm
3.76263
1.5
Table 4.28. Stress
Name
Type
Min
Max
Stress1
VON: von Mises Stress
0.0427868 N/mm^2
(MPa)
Node: 328
115.585 N/mm^2
(MPa)
Node: 4039
Part10-Study 1-Stress-Stress1
Table 4.29. Displacement
Name
Type
Min
Max
Displacement1
URES: Resultant Displacement
0 mm
Node: 5825
0.0660202 mm
Node: 97138
Part10-Study 1-Displacement-Displacement1
26
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Table 4.30. Strain
Name
Type
Min
Max
Strain1
ESTRN: Equivalent Strain
8.10288e-008
Element: 53984
0.000460036
Element: 37647
Part10-Study 1-Strain-Strain1
Table 4.31. Factor Safety
Name
Type
Min
Max
Factor of Safety1
Max von Mises Stress
1.9085
Node: 4039
5155.66
Node: 328
Part10-Study 1-Factor of Safety-Factor of Safety1
27
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
4C. Eccentric Shaft Results
Table 4.32. Model Information
Model name: shaft
Current Configuration: Default
Solid Bodies
Document Name and
Reference
Cut-Extrude3
Treated As
Volumetric Properties
Mass:0.236455 kg
Volume:3.03147e-005 m^3
Density:7800 kg/m^3
Weight:2.31726 N
Solid Body
28
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Document Path/Date
Modified
C:\Users\DELLi7\Desktop\
Solid\shaft.SLDPRT
May 23 18:36:44 2012
Table 4.33. Study Properties
Study name
Study 1
Analysis type
Static
Mesh type
Solid Mesh
Thermal Effect:
On
Thermal option
Include temperature loads
Zero strain temperature
298 Kelvin
Include fluid pressure effects from SolidWorks
Flow Simulation
Solver type
Off
Inplane Effect:
Off
Soft Spring:
Off
Inertial Relief:
Off
Incompatible bonding options
Automatic
Large displacement
Off
Compute free body forces
On
Friction
Off
Use Adaptive Method:
Off
Result folder
SolidWorks document
(C:\Users\DELLi7\Desktop\Solid)
FFEPlus
Table 4.34. Units
Unit system:
SI (MKS)
Length/Displacement
mm
Temperature
Kelvin
Angular velocity
Rad/sec
Pressure/Stress
N/m^2
29
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Table 4.35. Material Properties
Model Reference
Properties
Name:
Model type:
Default failure criterion:
Yield strength:
Tensile strength:
Elastic modulus:
Poisson's ratio:
Mass density:
Shear modulus:
Thermal expansion
coefficient:
Components
Plain Carbon Steel
Linear Elastic Isotropic
Unknown
2.20594e+008 N/m^2
3.99826e+008 N/m^2
2.1e+011 N/m^2
0.28
7800 kg/m^3
7.9e+010 N/m^2
1.3e-005 /Kelvin
SolidBody 1(CutExtrude3)(shaft)
Curve Data:N/A
Table 4.36. Loads and Fixtures
Fixture name
Fixture Image
Fixture Details
Entities:
Type:
1 face(s)
Fixed Geometry
Fixed-1
Resultant Forces
Components
Reaction force(N)
Reaction Moment(N-m)
Load name
X
0.101737
0
Y
-0.0974712
0
Load Image
Z
0.0701776
0
Resultant
0.157404
0
Load Details
Entities:
Reference:
Type:
Value:
Torque-1
30
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
1 face(s)
Face< 1 >
Apply torque
36 N-m
Table 4.37. Mesh Information
Mesh type
Solid Mesh
Mesher Used:
Curvature based mesh
Jacobian points
4 Points
Maximum element size
1.55957 mm
Minimum element size
0.519852 mm
Mesh Quality
High
Total Nodes
288067
Total Elements
199071
Maximum Aspect Ratio
4.6466
% of elements with Aspect Ratio < 3
99.8
% of elements with Aspect Ratio > 10
0
% of distorted elements(Jacobian)
0
Time to complete mesh(hh;mm;ss):
00:00:11
Computer name:
DELLI78GB
31
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Mesh Control Name
Mesh Control Image
Mesh Control Details
Entities:
116 edge(s), 2
face(s)
Units: mm
Size: 0.779785
Ratio: 1.5
Control-1
Table 4.38. Stress
Name
Type
Min
Max
Stress1
VON: von Mises Stress
0.0115002 N/mm^2 (MPa)
Node: 69002
147.131 N/mm^2 (MPa)
Node: 39057
shaft-Study 1-Stress-Stress1
32
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Table 4.39. Displacement
Name
Type
Min
Max
Displacement1
URES: Resultant Displacement
0 mm
Node: 259
0.0272518 mm
Node: 518
shaft-Study 1-Displacement-Displacement1
Table 4.40. Strain
Name
Type
Min
Max
Strain1
ESTRN: Equivalent Strain
1.00763e-006
Element: 7928
0.000403577
Element: 19615
shaft-Study 1-Strain-Strain1
33
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Table 4.41. Table Factor of Safety
Name
Type
Min
Max
Factor of Safety1
Max von Mises Stress
1.4993
Node: 39057
19181.7
Node: 69002
shaft-Study 1-Factor of Safety-Factor of Safety1
34
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
Appendix 4 - Drawings
DWG No. 1/5 – Gasket
DWG No. 2/5 – Cycloid Wheel
DWG No. 3/5 – Pin
DWG No. 4/5 – Housing
DWG No. 5/5 – Cycloid Stage
35
Biser Borislavov, Ivaylo Borisov, Vilislav Panchev
A
10
A
6.1° ±0.1°
28
295
10
5.5 ±0.01
10
0
59x
A-A
1:2
325
UNLESS OTHERWISE SPECIFIED:
DIMENSIONS ARE IN MILLIMETERS
SURFACE FINISH:
TOLERANCES:
LINEAR:
ANGULAR:
FINISH:
DEBUR AND
BREAK SHARP
EDGES
10
DO NOT SCALE DRAWING
Linnaeus University, Vaxjo
ISO 2768-mH
NAME
SIGNATURE
REVISION
DATE
TITLE:
Gasket
DRAWN
CHK'D
APPV'D
MFG
Q.A
MATERIAL:
Cast Alloy Steel
WEIGHT:
0,67 kg
DWG NO.
SCALE:1:5
A3
1/5
SHEET 1 OF 1
17
R54
.250
.75
0±
60
°
0.01
0.01 A
0.0
25
±0.0
25
0.8
(
)
0.01
53.500 ±0.015
6x
20
R1
10
15 ±0.01
xR
12
A
0.01 A
A
0.8
172 ±0
+0.02
288.50 - 0.03
53 ±0.3
.010
63.5 H7
6.3
0.8
60
°±
0.
20° ±0.01
01
°
°
A
Module
Number of teeth
Modification
Pitch diameter
Addendum diameter
Dedendum diameter
A-A
1:2
UNLESS OTHERWISE SPECIFIED:
DIMENSIONS ARE IN
MILLIMETERS SURFACE FINISH:
TOLERANCES: ISO 2768-mH
LINEAR:
ANGULAR:
NAME
DEBUR AND
BREAK SHARP
EDGES
FINISH:
SIGNATURE
DATE
DO NOT SCALE DRAWING
TITLE:
DRAWN
CHK'D
APPV'D
m
z
x
d1
da1
df1
5
58
0.35
290
288.5
281.5
REVISION
Cycloid Wheel
MFG
Q.A
MATERIAL:
20 MnCr 6
WEIGHT:
5,03 kg
DWG NO.
SCALE:1:5
A3
2/5
SHEET 1 OF 1
10
( )
10 ±0.025
46 ±0.02
0.8
0.01
A
1x45
0.01 A
UNLESS OTHERWISE SPECIFIED:
DIMENSIONS ARE IN
MILLIMETERS SURFACE FINISH:
TOLERANCES:ISO 2768-mH
LINEAR:
ANGULAR:
NAME
DEBUR AND
BREAK SHARP
EDGES
FINISH:
DO NOT SCALE DRAWING
REVISION
Linnaeus University, Vaxjo
SIGNATURE
DATE
TITLE:
Pin
DRAWN
CHK'D
APPV'D
MFG
Q.A
MATERIAL:
Plain Carbon Steel
WEIGHT:
0.028 kg
DWG NO.
SCALE:2:1
A4
3/5
SHEET 1 OF 1
A
45°
12
10
±0.
1°
(
)
(8x
)
x
1.8
00
59
0±
0.1
325
35
390
A
40
R5
36
8x
295 ±
A-A
1:2
0.05
UNLESS OTHERWISE SPECIFIED:
DIMENSIONS ARE IN
MILLIMETERS SURFACE FINISH:
TOLERANCES:ISO 2768-mH
LINEAR:
ANGULAR:
NAME
DEBUR AND
BREAK SHARP
EDGES
FINISH:
SIGNATURE
DATE
DO NOT SCALE DRAWING
REVISION
Linnaeus University, Vaxjo
TITLE:
Housing
DRAWN
CHK'D
APPV'D
MFG
Q.A
MATERIAL:
Cast Alloy Steel
WEIGHT:
13,5 kg
DWG NO.
SCALE:1:5
4/5
A3
SHEET 1 OF 1
4
46 ±0.1
A
1
3
2
A
390
35 ±0.1
A-A
4
3
2
1
Housing
Pin x59
Cycloid Wheel
Gasket
UNLESS OTHERWISE SPECIFIED:
DIMENSIONS ARE IN MILLIMETERS
SURFACE FINISH:
TOLERANCES:
LINEAR:
ANGULAR:
NAME
DEBUR AND
BREAK SHARP
EDGES
FINISH:
SIGNATURE
DATE
- DWG NO. 4/5
- DWG NO. 3/5
- DWG NO. 2/5
- DWG NO. 1/5
DO NOT SCALE DRAWING
REVISION
Linnaeus Univesity,Vaxjo
TITLE:
Cycloid stage
DRAWN
CHK'D
APPV'D
MFG
Q.A
MATERIAL:
WEIGHT:
DWG NO.
A2
5/5
SCALE 1:2
SHEET 1 OF 1